Classification of simple finite-dim Lie algebras for char >=5 has been accomplished not so long time ago, and char p=2,3 is open problem.

I wonder what is known/expected for char p=2,3 ?

More vague and soft question is the following - look at some famous classification problems: simple finite-dim Lie algebras, simple finite groups, some other things classified by ADE... We see the following pattern: there are some series of objects and finite number of "sporadic" objects. I.e. it never happens that there is infinite number of examples which are not in "series". So classification of simple objects is simple (in some very informal sense).

The question: can we expect this in advance, without obtaining classification ? (What are other examples/counter examples of similar phenomenon ?).

For example can we expect/prove this for simple Lie algs for char =2,3 ? I.e. there will be some finite number of series and finite number of "sporadic" examples ?

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    $\begingroup$ Arguably, prime numbers do not form an infinite family in any sort of simple way. If you agree with that, then the classification of finite fields, finite simple groups, and some other things all fail to be simple. $\endgroup$ – Will Sawin Nov 17 '12 at 22:23
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    $\begingroup$ It is not clear to me that "series plus sporadic" is a property of a classification; it seems to me more like a property of a particular way of looking at a classification, which is dependent on the current knowledge, habits, tastes, etc. of specific groups of mathematicians. $\endgroup$ – Qiaochu Yuan Nov 18 '12 at 1:22
  • $\begingroup$ @Qiaochu Yuan nevertheless we can say that "complexity" of such description is small - i.e. a code which will give for each natural "n" generators and relations of n-th simple Lie algebra (choose reasonable enumeration) will be quite short. (Kolmogorov's complexity is length of code to describe the object - as far as I understand). I am not sure it is equivalent to "series+sporadic", nevertheless it is measure of simplicity. $\endgroup$ – Alexander Chervov Nov 18 '12 at 6:44
  • $\begingroup$ @Qiaochu, as a general rule I would agree with you. However I think the Classification of Finite Simple Groups really does exhibit some kind of inherent series+sporadic behaviour. I wouldn't know how to formalise this - that might be a job for a logician (??) - but equally I don't think it can be explained away as a consequence of current mathematical culture. $\endgroup$ – Nick Gill Nov 19 '12 at 14:21
  • $\begingroup$ There seems to be some dividing line where tightly constrained cases have a series/sporadic structure, and relaxing the constraint too much destroys this. For example, the classification of finite simple Moufang loops adds one additional series to the list of finite simple groups, while relaxing this slightly to Bol loops destroys any chances of classification. $\endgroup$ – arsmath May 10 '13 at 16:51

According with the introduction of Strade's book "Simple Lie algebras over fields of positive characteristic. Structure Theory", it seems that a possible list of known finite-dimensional simple Lie algebras over algebraically closed fields of characteristic 3 could be close to complete. A discussion on this topics can be found in Section 4.4 of that book. On the other hand, in characteristic 2 the situation seems to be more complicated. For example, in the paper [Yu. Kochetov - D. Leites: Simple Lie algebras of characteristic 2 recovered from superalgebras and on the notion of a simple group, in Proceedings of the International Algebraic Conference in the Memory of A.I. Malcev, Novosibirsk, Contemp. Math. 131 (1992), 59-67] the authors have constructed simple Lie algebras in characteristic 2 from superalgebras. Thus one expects that a greater variety of constructions could get many more examples in this exceptional characteristic.


(This is too long for a comment). There is a recent surge of activity around (attempts of) classification of simple finite-dimensional Lie algebras in $p=2,3$. It is my understanding that the common view among experts is that the case $p=3$ might be in sight, while situation in $p=2$ is still chaotic. The main current players in the field are A. Grishkov, M.I. Kuznetsov and his students, and D. Leites and his collaborators.

As in Kostrikin-Shafarevich program for large characteristics, deformations (of some initial set of algebras) play a role. Computers are involved a lot.

A sample of recent publications:

  • S. Bouarroudj, P. Grozman, D. Leites, Infinitesimal deformations of symmetric simple modular Lie algebras and Lie superalgebras, arXiv:0807.3054
  • S. Bouarroudj, A. Lebedev, D. Leites, I. Shchepochkina, Deforms of Lie algebras in characteristic 2: semi-trivial for Jurman algebras, non-trivial for Kaplansky algebras, arXiv:1301.2781.
  • B. Eick, Some new simple Lie algebras in characteristic 2, J. Symb. Comput. 45 (2010), N9, 943-951
  • A. Grishkov, On simple Lie algebras over a field of characteristic 2, J. Algebra 363 (2012), 14-18
  • A. Grishkov and M. Guerreiro, On simple Lie algebras of dimension seven over fields of characteristic 2, Sao Paulo J. Math. Sci. 4 (2010), N1, 93--107
  • D. Leites, Towards classification of simple finite dimensional modular Lie superalgebras, arXiv:0710.5638.
  • M. Vaughan-Lee, Simple Lie algebras of low dimension over GF(2), LMS J. Comput. Math. 9 (2006), 174--192

In general, the classification of finite-dimensional simple non-associative algebras is difficult, i.e., not known, even for algebraically closed fields of characteristic zero. I have tried to start a classification of all complex simple pre-Lie algebras. The first results show that a classification will be very difficult (see http://homepage.univie.ac.at/Dietrich.Burde/papers/burde_08_simple_lsa.pdf). The special case of complex simple Novikov algebras has been solved by E. Zelmanov.


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